if it makes you feel any better, time travel has been proven by experiment, although it was just a few nanoseconds...
they got 2 really accurate clocks in sync and put 1 in a jet and flew it around really fast. Then they compared the clocks and they were no longer in sync.

it wasn't really time travel per se, it was just messing with time. There's nothing dangerous about it because you can only "travel into the future" so there's no way to screw with things... en.wikipedia.org/wiki/Time_dilation

Well how about you only need to be traveling really close to the speed of light to actually get somewhere really fast. When you approach the speed of light, time from your point of view slows down, so you could travel lightyears in a few minutes, assuming you had the energy. But from everyone else's point of view, the trip would take the normal time.